Most people wondering how does direction reversal occur including me, but I guess it could be a natural consequence of dividing unit motion by two reciprocal aspect which is implied by your notation S|T. One familiar with music would understand it is like a string of one unit cut into 1/2 and extend to 2/1. The SUDR and TUDR both have a frequency(1/t) of 1/2, S|T 1/1 and 2S|T and S|2T have a frequency of 3/2. Am I correct?

Thanks for your question Sun. Sorry it has taken so long to respond. The reversals are definitely a philosophical challenge, but assuming them, we derive the natural numbers from the integers and a whole slew of other things.

Starting with a unit ratio of space/time change (s/t = 1/1), the first possibility is 1/2 and 2/1, depending upon which aspect is reversing. These two ratios equate or resolve to -1 and +1 respectively and the combinations of them can be used to arrive at any natural number.

These fundamental units of motion were called time and space "displacements" in Larson's works.

At the LRC, we call them SUDRs and TUDRs, denoting them with upper case S and T. A combination of them is denoted S|T.

Combining 1/2 and 2/1 on a calculator generates a sum of 2.5, but this is due to the underlying assumption that reciprocals of positive integers should be treated as fractions of a whole- instead of reciprocal integers.

As far as frequencies go, there is one cycle of reversal per two units of time (space). Summing frequencies, however, is trickier than summing displacements. As you probably know, combining two radio frequencies produces both the sum and difference frequencies, but the frequency of 2/1 is four times the frequency of 1/2.

Moreover, the absolute value of the space over time frequency (1/2) is equal to that of the time over space frequency (2/1). Taking this seeming contradiction into account requires us to think about it a little differently, which I'll try to explain directly.

Hello Doug,
Thank you for your presentation.
Let me use a notation of a-c-b for my own convenient to represent your equation.
Am i correct that you assume everything starts from one net displacement, 1/2 and 2/1? Particles are consequences that combine variable numbers of 1/2 and 2/1 with variable numbers of 1/1? 1/1 represent for unit motion? a, c, b stand for the each dimension of motion?
How did you get 2S|T = 2/4 + 2/1 + 2/1? Why it is not S|T+SUDR=1/2+1/1+2/1+2/1?

The basic S|T equation, S|T = 1/2 + 1/1 + 2/1 = 4|4 num, expresses the total scalar motion of the SUDR (s/ t = 1/2) and TUDR (s/t = 2/1) combination. The middle term (1/1) consists of the inner portion of the SUDR oscillation (the numerator) and the inner portion of the TUDR oscillation (the denominator.) Without it the total of natural units of motion (num), would be incorrect.

The ratio of SUDRs to TUDRs (Ss and Ts) in this equation is 1:1, or balanced. We can unbalance it in two "directions," by adding an S or a T to the equation. Adding an S, S|T + S, gives us 2S|T. Adding a T, gives us S|2T.

I hope that make more sense now. This new math is a little tricky, because not only do we add numerators to numerators and denominators to denominators, but we CONSTRUCT the middle term from the LH (numerator) and the RH (denominator) terms, after the fact so-to-speak.